The logarithmic Minkowski problem for polytopes

The logarithmic Minkowski problem asks for necessary and sufficient conditions for a finite Borel measure on the unit sphere so that it is the cone-volume measure of a convex body. This problem was solved recently by Böröczky, Lutwak, Yang and Zhang for even measures (Böröczky et al. (2013) [8]). This paper solves the case of discrete measures whose supports are in general position.